Specialization to the tangent cone and Whitney equisingularity (Q2861488)

Let \((X,0)\) be a reduced, equidimensional germ of an analytic singularity with reduced tangent cone \((C_{X,0},0)\). Let \(I\subseteq \mathbb{C}\{z_0, \dots, z_n\}\) be the ideal defining \((X,0)\) and \(f_1, \dots, f_k\) be generators of \(I\) such that their initial forms generate the initial ideal defining the tangent cone. Let \(m_i\) be the degree of the initial form of \(f_i\) and \(F_i(z_0, \dots, z_n, t)=t^{-m_i} f_i(tz_0, \dots, tz_n)\). Let \(J=\langle f_1, \dots, f_k\rangle\).NEWLINENEWLINELet \(\varphi:(\mathfrak{X},0)\to (\mathbb{C},0)\) be the flat family with section \(\sigma\) defined by \(J\). Then for each \(t\in \mathbb{C}\smallsetminus\{0\}\) the germ \((\varphi^{-1}(t), \sigma(t))\) is isomorphic to \((X, 0)\) and the special fibre is isomorphic to the tangent cone. Let \(Y\subset \mathfrak{X}\) be defined by the section and \(\mathfrak{X}^0\) be the non singular part of \(\mathfrak{X}\). it is proved that the couple \((\mathfrak{X}^0, Y)\) satisfies Whitney's condition a) and b) at the origin if \((X,0)\) does not have exceptional cones and the tangent cone is reduced.